Geo-Metric-Affine-Projective Computing
|
|
- Oscar Gibbs
- 5 years ago
- Views:
Transcription
1 Geo-Metric-Affine-Projective Computing Speaker: Ambjörn Naeve, Ph.D., Senior researcher Affiliation: Centre for user-oriented IT-Design (CID) Dept. of Numerical Analysis and Computing Science Royal Institute of Technology (KTH) Stockholm, Sweden -address: web-site: cid.nada.kth.se/il
2 Projective Drawing Board (PDB) PDB is an interactive program for doing plane projective geometry that will be used to illustrate this lecture. PDB has been developed by Harald Winroth and Ambjörn Naeve as a part of Harald s doctoral thesis work at the Computational Vision and Active Perception (CVAP) laboratory at KTH. PDB is avaliable as freeware under Linux.
3
4
5 Geometric algebra in n-dim Euclidean space Underlying vector space V n with ON-basis e 1,...,e n. Geometric algebra: G= G G( V n ) has 2 n dimensions. A multivector is a sum of k-vectors: A k-blade = blade of grade k : B = b b b n k M n = k = 0 M k A k-vector is a sum of k-blades: M = A + B +K k k k Note: B 0 b, K, b 1 Hence: k k 1 2 K are linearly independent. the grade of a blade is the dimension of the subspace that it spans. k
6 Blades correspond to geometric objects blade of grade equivalence class of directed equal orientation and 1 line segments 2 surface regions 3 3-dim regions length area volume k k-dim regions k-volume
7 Pseudoscalars and duality Def: A n-blade in G n is called a pseudoscalar. A pseudoscalar: P = p p K p 1 2 n A unit pseudoscalar: I = e e K e 1 2 n The bracket of P: [ P]= PI 1 The dual of a multivector x: Dual( x) = xi 1 Notation: Note: Dual( x) If A is a k-blade, then = A * x * is a (n-k)-blade.
8 The subspace of a blade Fact: Fact: To every non-zero m-blade B = b K b 1 m there corresponds a m-dim subspace B n V with If B= Linspan{ b, K, b } 1 m e, e, K, is an ON-basis for B 1 2 e m m and if b = b e = = Linspan{ b V : b B= 0 }. i k 1 ik k then B= ( det b ) e e K e 1 2 ik = ( ) det b ee ik n for i = 1,,m, e K 1 2 m. m
9 Dual subspaces <=> orthogonal complements Fact: If A is a non-zero m-blade A * = A. Proof: We can WLOG choose an ON-basis for V n such that A= λee Ke 1 2 m We then have A * = AI 1 =± λem+ which implies that and I = ee Ke 1 2 n. 1 K e n A * = A.
10 The join and the meet of two blades Def: Given blades A and B, if there exists a blade C such that A= BC = B C we say that A is a dividend of B and B is a divisor of A. Def: The join of blades A and B is a common dividend of lowest grade. Def: The meet of blades A and B is a common divisor of greatest grade. The join and meet provide a representation in geometric algebra of the lattice algebra of subspaces of V n.
11 Join of two blades <=> sum of their subspaces Def: For two blades A and B with A B 0 Join( AB, ) = A B we can define:. In this case:. Example: 0 A B= A+ B and A B=0 ab, V 3 a b a b a b =0 a b = a + b = { λ a+ µ b: λ, µ R}
12 Meet of blades <=> intersection of subspaces Def: If blades AB then:, 0 and A + B =V * * Meet( AB, ) A B= ( A B) I. In this case: A B= A B. n Note: The meet product is related to the outer product by duality: ( A B) I 1 = ( A B ) II 1 = A B * * * * ( A B) = ( A) ( B) Dual Dual Dual
13 Dual outer product Dualisation: G G x a x * = xi * Dual outer product: 1 G G G * * * G G G x y = (( xi ) ( yi )) I * * * x y = ( x y) 1 1
14 Example: * ( ) A = e e I * 1 2 ( ) B = e e I 2 3 A B= ( A B ) I V 3, I = e e e = eee A= e e = ee, B= e e = ee * * = ( ee ) ( eee) = ( ) = ( ee) ( eee) = e ( ) = e e ( eee ) = e eeeee = e 3 A * = e 3 B Hence: A B= A B e 2 e 1 = A A = B * B
15 Projective geometry - historical perspective 1-d subspace parallell to the ground plane line at infinity horizon point at infinity eye parallell lines point ground plane 1-dim subspace through the eye
16 n-dimensional projective space P n P n = P( V n+1) A point p is a 1-dim subspace p = the set of non-zero subspaces of V n+1. (spanned by a 1-blade a). = a = { λa : λ 0} = a = α, α a 0. A line l is a 2-dim subspace (spanned by a 2-blade B 2 ). l = B = { λb : λ 0} 2 2 = B 2. Let B denote the set of non-zero blades of the geometric algebra G(V n+1 ). B B P n. Hence we have the mapping B
17 The projective plane P 2 0 abx,, V 3 Line: a b P 2 { x 0: x a= 0} a b { x 0: x a b= 0}
18 The intersection of two lines in P 2 AB, {non-zero 2-blades in G } 3. P 2 A B B A
19 Collinear points p q p q r The points p, q, r are collinear if and only if p q r =0.
20 Concurrent lines P R P The lines P,, R are concurrent if and only if ( P ) R=0.
21 Desargues configuration a a b b = b b c c q p A= b c A = b c B= c a B = c a p= A A q= B B C = a b r r = C C C = a b P= a a R= c c
22 Desargues configuration (cont.) P= a a = b b R= c c p= A A q = B B r = C C = ( b c) ( b c ) = ( c a) ( c a ) = ( a b) ( a b ) J = a b c =[ abc] I J = a b c = [ abc ] I Leads to: p q r JJ ( P ) R = 0 if and only if = 0 if and only if p,q,r are collinear P,,R are concurrent
23 Desargues theorem B P C A B A C L R p, q, r are collinear iff P,, R are concurrent.
24 Pascal s theorem Let abca,,,, b be five given points in P 2 Consider the second degree polynomial given by px ( ) = (( a b ) ( a b)) (( b x) ( b c)) (( c a ) ( x a)).. It is obvious that pa ( ) = pb ( ) =0 and easy to verify that pc () = pa ( ) = pb ( ) = 0. Hence: px ( )= 0 must be the equation of the conic on the 5 given points.
25 Verifying that p(a ) = 0 pa ( ) = same line = (( a b ) ( a b)) (( b a ) ( b c)) same point (( c a ) ( a a)) = a b = ( a b) a = 0 = a
26 Pascal s theorem (cont.) Hence, a sixth point c lies on this conic if and only if pc ( ) = (( a b ) ( a b)) (( b c ) ( b c)) (( c a ) ( c a)) =0. Geometric formulation: The three points of intersection of opposite sides of a hexagon inscribed in a conic are collinear. This is a property of the hexagrammum mysticum, which Blaise Pascal discovered in 1640, at the age of 16.
27 Pascal s theorem (cont.) a a 4 1 r b t 5 6 b 2 r = ( a b ) ( a b) s= ( b c ) ( b c) t = ( c a ) ( c a) s 3 c c = = = 1 4 r s t =
28 Pascal s theorem (cont.) L P P R R The three points of intersection of opposite sides of a hexagon inscribed in a conic are collinear.
29 Pappus theorem (ca 350 A.D.) R P P R L If the conic degenerates into two straight lines, Pappus theorem emerges as a special case of Pascal s.
30 Outermorphisms Def: A mapping f : G G is called an outermorphism if (i) f is linear. (ii) f()= t t t R (iii) f( x y) = f( x) f( y) xy, G (iv) f k ( G ) G k k 0
31 The induced outermorphism Let Fact: T : V V denote a linear mapping. T induces an outermorphism T : G G given by Ta ( K a) = Ta ( ) K Ta ( ) T( λ) 1 k 1 = λ and linear extension., λ R k Interpretation: T maps the blades of V in accordance with how T maps the vectors of V.
32 Polarization with respect to a quadric in P n Let T n+ 1 n+ 1 : V V denote a symmetric linear map, which means that Tx ( ) y= xty ( ) xy, n+ V 1. The corresponding quadric (hyper)surface in is given by n+ = { x V 1 : x T( x) = 0, x 0 }., P n Def: The polar of the k-blade A with respect to is the (n+1-k)-blade defined by Pol ( A) T( A) * T( A) I 1.
33 Polarization (cont.) Note: T = id In this case = { x V 1 : x x = 0, x 0 }. 1 Pol ( A) AI A and polarization becomes identical to dualization. n+ * Fact: For a blade A we have (i) (ii) Pol ( Pol ( A)) = A If A is tangent to then Pol (A) is tangent to. Especially: If x is a point on, then Pol (x) is the hyperplane which is tangent to at the point x.
34 Pol ( x) x
35 x Pol ( x)
36 Pol ( x) x
37 Polarity with respect to a conic P 2 y x y x Pol ( x) Pol ( y) Pol ( x) Pol ( y) The polar of the join of x and y is the meet of the polars of x and y = = Pol ( x y) Pol ( x) Pol ( y)
38 Pol ( x) Pol ( y) x x y y
39 Polar reciprocity Let xy, n+ V 1 represent two points in P n. Then we have from the symmetry of T : y T( x) * = y ( T( x) I 1 ) = ( ytx ( )) I 1 = ( x T( y)) I 1 = x 1 ( T( y) I ) = x T( y) *. Hence: y Pol ( x) = 0 i.e. the point y lies on the polar of the point x if and only if x lies on the polar of y. x Pol ( y) = 0
40 Brianchon s theorem Pascal line (1640) 3 2 Brianchon point (1810)
41 The dual map Let f : G G and assume that be linear, I 2 0. G * * f G Def: The dual map f : G G is the linear map given by f ( x) = f( xi) I 1 G * * G f f ( x ) = f ~ f( x) = f G G Note: 1 f( x) = f ( xi ) I = = f ( xi 1 I 2 ) I 1 f ( xi 1 ) 2 2 I I I = f ( xi) I 1 ( = f x)
42 Polarizing a quadric with respect to another Let S : G G and T : G G be symmetric outermorphisms, Fact: P= { x G : x S( x) = 0, x 0} = { x G : x T( x) = 0, x 0} be the corresponding two quadrics. Polarizing the multivectors of the quadric P with respect to the quadric gives a quadric R with equation and we have x and let x ( ToS ot( x)) = 0 P Pol ( x) R,.
43 The reciprocal quadric Polarizing the quadric with respect to the quadric generates the quadric P : : x S( x) = 0 x T( x) = 0 R : x ( ToS ot( x)) = 0 R P
44 Cartesian-Affine-Projective relationships V n x V n+1 V n y The affine part of Projective Space: * x * x = x+ e A n n+ 1 y = ( ye ) 1 y e * n+ 1 well defined e n+1 y y P n V n P n V n n+ 1 V n A n * x ( ye ) n+ 1 1 y Affine space V n+1 V n e 1 x = y * * x * e n Cartesian space
45 The intersection of two lines in the plane a,b,c,d p * ( ) *( ) A,B,C,D P = (A B) (C D) A e 3 C P D B Affine plane R 2 a c p d b Cartesian plane
46 The intersection of two lines (cont.) P = ( A B) ( C D) [ ] [ ] = A B C D A B D C [ A B C ]=( A B C) I 1 0 = does not contain e 3 ( ) = ( a+ e ) ( b+ e ) ( c+ e ) eee ( ) = ( a c) ( b c) ( c+ e ) eee ( ) + = ( a c) ( b c) c eee ( ) (( ) ( ) ) = ( a c) ( b c) ee ( a c) ( b c) e eee = a c b c e eee = αd α βc
47 The intersection of two lines (cont.) In the same way we get β = [ A B D ] Hence we can write p as: p = * P = ( αd βc) * = ( A B D) I 1 = ( a d) ( b d) ee 2 1 = (( αd βc) e ) ( αd βc) e = ( α β) ( αd+ αe βc βe ) e 1 = ( α β) ( αd βc) does not contain e 3 does not contain e 3
48 Reflection in a plane-curve mirror s m(s) = = m m 1 t m(s+ds) = m(s). t 1 = t(s+ds) M p uu 1 = -t(p-m)t in-caustic curvature centre of the mirror r out-caustic = - t 1 (p-m 1 )t 1 q = lim ds (m u) (m 1 u 1 )
49 Reflection in a plane-curve mirror (cont.) Making use of the intersection formula deduced earlier and introducing we get t n = t for the unit mirror normal q m= (( p m) t) t ( p m) n) n 1 2 m p ( m) ( p m) n 2 This is an expression of Tschirnhausen s reflection law.
50 Reflection in a plane-curve mirror (cont.) m Tschirnhausens reflection formula p r u v q ± = u m q m v m
51 References: Hestenes, D. & Ziegler, R., Projective Geometry with Clifford Algebra, Acta Applicandae Mathematicae 23, pp , Naeve, A. & Svensson, L., Geo-Metric-Affine-Projective Unification, Sommer (ed.), Geometric Computations with Clifford Algebra, Chapter 5, pp , Springer, Winroth, H., Dynamic Projective Geometry, TRITA-NA-99/01, ISSN , ISRN KTH/NA/R--99/01--SE, Dissertation, The Computational Vision and Active Perception Laboratory, Dept. of Numerical Analysis and Computing Science, KTH, Stockholm, March 1999.
The geometry of projective space
Chapter 1 The geometry of projective space 1.1 Projective spaces Definition. A vector subspace of a vector space V is a non-empty subset U V which is closed under addition and scalar multiplication. In
More informationAutomated Geometric Reasoning with Geometric Algebra: Practice and Theory
Automated Geometric Reasoning with Geometric Algebra: Practice and Theory Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing 2017.07.25
More informationProjective Geometry with Clifford Algebra*
In: Acta Applicandae Mathematicae, Vol. 23, (1991) 25 63. Projective Geometry with Clifford Algebra* DAVID HESTENES and RENATUS ZIEGLER Abstract. Projective geometry is formulated in the language of geometric
More informationConics. Chapter 6. When you do this Exercise, you will see that the intersections are of the form. ax 2 + bxy + cy 2 + ex + fy + g =0, (1)
Chapter 6 Conics We have a pretty good understanding of lines/linear equations in R 2 and RP 2. Let s spend some time studying quadratic equations in these spaces. These curves are called conics for the
More informationarxiv: v6 [math.mg] 9 May 2014
arxiv:1311.0131v6 [math.mg] 9 May 2014 A Clifford algebraic Approach to Line Geometry Daniel Klawitter Abstract. In this paper we combine methods from projective geometry, Klein s model, and Clifford algebra.
More informationTHE THEOREM OF DESARGUES
THE THEOREM OF DESARGUES TIMOTHY VIS 1. Introduction In this worksheet we will be exploring some proofs surrounding the Theorem of Desargues. This theorem plays an extremely important role in projective
More informationHyperbolic Conformal Geometry with Clifford Algebra 1)
MM Research Preprints, 72 83 No. 18, Dec. 1999. Beijing Hyperbolic Conformal Geometry with Clifford Algebra 1) Hongbo Li Abstract. In this paper we study hyperbolic conformal geometry following a Clifford
More informationarxiv:math/ v1 [math.ag] 3 Mar 2002
How to sum up triangles arxiv:math/0203022v1 [math.ag] 3 Mar 2002 Bakharev F. Kokhas K. Petrov F. June 2001 Abstract We prove configuration theorems that generalize the Desargues, Pascal, and Pappus theorems.
More informationCLIFFORD GEOMETRIC ALGEBRAS IN MULTILINEAR ALGEBRA AND NON-EUCLIDEAN GEOMETRIES
CLIFFORD GEOMETRIC ALGEBRAS IN MULTILINEAR ALGEBRA AND NON-EUCLIDEAN GEOMETRIES Garret Sobczyk Universidad de las Américas Departamento de Físico-Matemáticas Apartado Postal #100, Santa Catarina Mártir
More informationConics and their duals
9 Conics and their duals You always admire what you really don t understand. Blaise Pascal So far we dealt almost exclusively with situations in which only points and lines were involved. Geometry would
More informationLecture 3: Geometry. Sadov s theorem. Morley s miracle. The butterfly theorem. E-320: Teaching Math with a Historical Perspective Oliver Knill, 2013
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2013 Lecture 3: Geometry Sadov s theorem Here is an example of a geometric theorem which has been found only 10 years ago: The quadrilateral
More informationTopic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths
Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is
More information16. Analysis and Computation of the Intrinsic Camera Parameters
16. Analysis and Computation of the Intrinsic Camera Parameters Eduardo Bayro-Corrochano and Bodo Rosenhahn Institute of Computer Science and Applied Mathematics, Christian-Albrechts-University of Kiel
More informationChapter 1 Numerical evaluation of Versors with Clifford Algebra
This is page 1 Printer: Opaque this Chapter 1 Numerical evaluation of Versors with Clifford Algebra Christian B. U. Perwass, Gerald Sommer 1 ABSTRACT This paper has two main parts. In the first part we
More informationCollinearity/Concurrence
Collinearity/Concurrence Ray Li (rayyli@stanford.edu) June 29, 2017 1 Introduction/Facts you should know 1. (Cevian Triangle) Let ABC be a triangle and P be a point. Let lines AP, BP, CP meet lines BC,
More informationConics and perspectivities
10 Conics and perspectivities Some people have got a mental horizon of radius zero and call it their point of view. Attributed to David Hilbert In the last chapter we treated conics more or less as isolated
More informationTwo applications of the theorem of Carnot
Annales Mathematicae et Informaticae 40 (2012) pp. 135 144 http://ami.ektf.hu Two applications of the theorem of Carnot Zoltán Szilasi Institute of Mathematics, MTA-DE Research Group Equations, Functions
More informationOld Wine in New Bottles: A new algebraic framework for computational geometry
Old Wine in New Bottles: A new algebraic framework for computational geometry 1 Introduction David Hestenes My purpose in this chapter is to introduce you to a powerful new algebraic model for Euclidean
More informationOn Pappus configuration in non-commutative projective geometry
Linear Algebra and its Applications 331 (2001) 203 209 www.elsevier.com/locate/laa On Pappus configuration in non-commutative projective geometry Giorgio Donati Dipartimento di Matematica della, Seconda
More informationEuclidean projective geometry: reciprocal polar varieties and focal loci
Euclidean projective geometry: reciprocal polar varieties and focal loci Ragni Piene Boris 60 Stockholm University June 1, 2017 Pole and polars in the plane Let Q be a conic section. Let P be a point in
More informationDefinition: A vector is a directed line segment which represents a displacement from one point P to another point Q.
THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH Algebra Section : - Introduction to Vectors. You may have already met the notion of a vector in physics. There you will have
More informationTwo applications of the theorem of Carnot
Two applications of the theorem of Carnot Zoltán Szilasi Abstract Using the theorem of Carnot we give elementary proofs of two statements of C Bradley We prove his conjecture concerning the tangents to
More informationGeometric Algebra. 4. Algebraic Foundations and 4D. Dr Chris Doran ARM Research
Geometric Algebra 4. Algebraic Foundations and 4D Dr Chris Doran ARM Research L4 S2 Axioms Elements of a geometric algebra are called multivectors Multivectors can be classified by grade Grade-0 terms
More informationCGAlgebra: a Mathematica package for conformal geometric algebra
arxiv:1711.02513v2 [cs.ms] 23 Aug 2018 CGAlgebra: a Mathematica package for conformal geometric algebra José L. Aragón Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de
More informationdiv(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let:
Algebraic Curves/Fall 015 Aaron Bertram 4. Projective Plane Curves are hypersurfaces in the plane CP. When nonsingular, they are Riemann surfaces, but we will also consider plane curves with singularities.
More informationNew Foundations in Mathematics: The Geometric Concept of Number
New Foundations in Mathematics: The Geometric Concept of Number by Garret Sobczyk Universidad de Las Americas-P Cholula, Mexico November 2012 What is Geometric Algebra? Geometric algebra is the completion
More informationA very simple proof of Pascal s hexagon theorem and some applications
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 5, November 2010, pp. 619 629. Indian Academy of Sciences A very simple proof of Pascal s hexagon theorem and some applications NEDELJKO STEFANOVIĆ and
More informationAn Elementary Proof of Poncelet s Theorem
An Elementary Proof of Poncelet s Theorem (on the occasion to its bicentennial) Lorenz Halbeisen (University of Zürich) Norbert Hungerbühler (ETH Zürich) Abstract We present an elementary proof of Poncelet
More informationAlgorithm for conversion between geometric algebra versor notation and conventional crystallographic symmetry-operation symbols
Algorithm for conversion between geometric algebra versor notation and conventional crystallographic symmetry-operation symbols Eckhard Hitzer and Christian Perwass June, 2009 Introduction This paper establishes
More informationR1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member
Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers
More informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More informationQUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)
QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents
More informationWhat Do Lattice Paths Have To Do With Matrices, And What Is Beyond Both?
What Do Lattice Paths Have To Do With Matrices, And What Is Beyond Both? Joseph E. Bonin The George Washington University These slides are available at blogs.gwu.edu/jbonin Some ideas in this talk were
More informationLocal properties of plane algebraic curves
Chapter 7 Local properties of plane algebraic curves Throughout this chapter let K be an algebraically closed field of characteristic zero, and as usual let A (K) be embedded into P (K) by identifying
More informationTHE GEOMETRY IN GEOMETRIC ALGEBRA THESIS. Presented to the Faculty. of the University of Alaska Fairbanks. in Partial Fulfillment of the Requirements
THE GEOMETRY IN GEOMETRIC ALGEBRA A THESIS Presented to the Faculty of the University of Alaska Fairbanks in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE By Kristopher N.
More informationarxiv: v1 [math-ph] 30 Jul 2009
Clifford algebra, geometric algebra, and applications Douglas Lundholm and Lars Svensson arxiv:0907.5356v1 [math-ph] 30 Jul 2009 Department of Mathematics, KTH SE-100 44 Stockholm, Sweden Abstract These
More informationSecant varieties. Marin Petkovic. November 23, 2015
Secant varieties Marin Petkovic November 23, 2015 Abstract The goal of this talk is to introduce secant varieies and show connections of secant varieties of Veronese variety to the Waring problem. 1 Secant
More informationPolar varieties and Euclidean distance degree
Polar varieties and Euclidean distance degree Ragni Piene SIAM AG17 Atlanta, Georgia, USA August 3, 2017 Pole and polars in the plane Let Q be a conic section. Let P be a point in the plane. There are
More informationConjugate conics and closed chains of Poncelet polygons
Mitt. Math. Ges. Hamburg?? (????), 1 0 Conjugate conics and closed chains of Poncelet polygons Lorenz Halbeisen Department of Mathematics, ETH Zentrum, Rämistrasse 101, 8092 Zürich, Switzerland Norbert
More information13 Spherical geometry
13 Spherical geometry Let ABC be a triangle in the Euclidean plane. From now on, we indicate the interior angles A = CAB, B = ABC, C = BCA at the vertices merely by A, B, C. The sides of length a = BC
More information6.1. Rational Expressions and Functions; Multiplying and Dividing. Copyright 2016, 2012, 2008 Pearson Education, Inc. 1
6.1 Rational Expressions and Functions; Multiplying and Dividing 1. Define rational expressions.. Define rational functions and give their domains. 3. Write rational expressions in lowest terms. 4. Multiply
More information2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time is
. If P(A) = x, P = 2x, P(A B) = 2, P ( A B) = 2 3, then the value of x is (A) 5 8 5 36 6 36 36 2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time
More informationConic Sections and Meet Intersections in Geometric Algebra
Conic Sections and Meet Intersections in Geometric Algebra Eckhard M.S. Hitzer Department of Physical Engineering, University of Fukui, Japan hitzer@mech.fukui-u.ac.jp Abstract. This paper first gives
More informationClassical Theorems in Plane Geometry 1
BERKELEY MATH CIRCLE 1999 2000 Classical Theorems in Plane Geometry 1 Zvezdelina Stankova-Frenkel UC Berkeley and Mills College Note: All objects in this handout are planar - i.e. they lie in the usual
More informationTHE ENVELOPE OF LINES MEETING A FIXED LINE AND TANGENT TO TWO SPHERES
6 September 2004 THE ENVELOPE OF LINES MEETING A FIXED LINE AND TANGENT TO TWO SPHERES Abstract. We study the set of lines that meet a fixed line and are tangent to two spheres and classify the configurations
More informationMath 3C Lecture 25. John Douglas Moore
Math 3C Lecture 25 John Douglas Moore June 1, 2009 Let V be a vector space. A basis for V is a collection of vectors {v 1,..., v k } such that 1. V = Span{v 1,..., v k }, and 2. {v 1,..., v k } are linearly
More informationTARGET : JEE 2013 SCORE. JEE (Advanced) Home Assignment # 03. Kota Chandigarh Ahmedabad
TARGT : J 01 SCOR J (Advanced) Home Assignment # 0 Kota Chandigarh Ahmedabad J-Mathematics HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP 1. If x + y = 0 is a tangent at the vertex of a parabola and x + y 7 =
More informationLecture: Linear algebra. 4. Solutions of linear equation systems The fundamental theorem of linear algebra
Lecture: Linear algebra. 1. Subspaces. 2. Orthogonal complement. 3. The four fundamental subspaces 4. Solutions of linear equation systems The fundamental theorem of linear algebra 5. Determining the fundamental
More informationUnless otherwise specified, V denotes an arbitrary finite-dimensional vector space.
MAT 90 // 0 points Exam Solutions Unless otherwise specified, V denotes an arbitrary finite-dimensional vector space..(0) Prove: a central arrangement A in V is essential if and only if the dual projective
More informationfive line proofs Victor Ufnarovski, Frank Wikström 20 december 2016 Matematikcentrum, LTH
five line proofs Victor Ufnarovski, Frank Wikström 20 december 2016 Matematikcentrum, LTH projections The orthogonal projections of a body on two different planes are disks. Prove that they have the same
More informationPROJECTIVE GEOMETRY. Contents
PROJECTIVE GEOMETRY Abstract. Projective geometry Contents 5. Geometric constructions, projective transformations, transitivity on triples, projective pl 1. Ceva, Menelaus 1 2. Desargues theorem, Pappus
More informationQ.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or
STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. A variable rectangle PQRS has its sides parallel to fied directions. Q and S lie respectivel on the lines = a, = a and P lies on the ais. Then the locus of R
More informationConcurrency and Collinearity
Concurrency and Collinearity Victoria Krakovna vkrakovna@gmail.com 1 Elementary Tools Here are some tips for concurrency and collinearity questions: 1. You can often restate a concurrency question as a
More informationD - E - F - G (1967 Jr.) Given that then find the number of real solutions ( ) of this equation.
D - E - F - G - 18 1. (1975 Jr.) Given and. Two circles, with centres and, touch each other and also the sides of the rectangle at and. If the radius of the smaller circle is 2, then find the radius of
More informationHomogeneous Coordinates
Homogeneous Coordinates Basilio Bona DAUIN-Politecnico di Torino October 2013 Basilio Bona (DAUIN-Politecnico di Torino) Homogeneous Coordinates October 2013 1 / 32 Introduction Homogeneous coordinates
More informationCHAPTER 10 VECTORS POINTS TO REMEMBER
For more important questions visit : www4onocom CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two
More informationThe structure tensor in projective spaces
The structure tensor in projective spaces Klas Nordberg Computer Vision Laboratory Department of Electrical Engineering Linköping University Sweden Abstract The structure tensor has been used mainly for
More informationProjection pencils of quadrics and Ivory s theorem
Projection pencils of quadrics and Ivory s theorem Á.G. Horváth Abstract. Using selfadjoint regular endomorphisms, the authors of [7] defined, for an indefinite inner product, a variant of the notion of
More informationDesign Theory Notes 1:
----------------------------------------------------------------------------------------------- Math 6023 Topics in Discrete Math: Design and Graph Theory Fall 2007 ------------------------------------------------------------------------------------------------
More informationChapter 8. Exploring Polynomial Functions. Jennifer Huss
Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial
More informationLattice Theory Lecture 4. Non-distributive lattices
Lattice Theory Lecture 4 Non-distributive lattices John Harding New Mexico State University www.math.nmsu.edu/ JohnHarding.html jharding@nmsu.edu Toulouse, July 2017 Introduction Here we mostly consider
More information( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear.
Problems 01 - POINT Page 1 ( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear. ( ) Prove that the two lines joining the mid-points of the pairs of opposite sides and the line
More informationPentagramma Mirificum old wine into new wineskins
Pentagramma Mirificum old wine into new wineskins R. Schwartz and S. Tabachnikov (arxiv:0910.1952; Math. Intelligencer, to appear) Lyon, November 2009, Claude s B-Day 1 Three classical theorems, Pappus
More informationGeometry JWR. Monday September 29, 2003
Geometry JWR Monday September 29, 2003 1 Foundations In this section we see how to view geometry as algebra. The ideas here should be familiar to the reader who has learned some analytic geometry (including
More informationMODEL ANSWERS TO HWK #3
MODEL ANSWERS TO HWK #3 1. Suppose that the point p = [v] and that the plane H corresponds to W V. Then a line l containing p, contained in H is spanned by the vector v and a vector w W, so that as a point
More informationDUALITY AND INSCRIBED ELLIPSES
DUALITY AND INSCRIBED ELLIPSES MAHESH AGARWAL, JOHN CLIFFORD, AND MICHAEL LACHANCE Abstract. We give a constructive proof for the existence of inscribed family of ellipses in convex n-gons for 3 n 5 using
More information(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2
CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5
More informationSURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS
SURA's Guides for rd to 1th Std for all Subjects in TM & EM Available 10 th STD. MARCH - 017 Public Exam Question Paper with Answers MATHEMATICS [Time Allowed : ½ Hrs.] [Maximum Marks : 100] SECTION -
More informationVector Spaces and Projective Geometry
Vector Spaces and Projective Geometry Lou de Boer December 20, 2009 Vector spaces are of fundamental importance, not only in Mathematics, but in many applied sciences as well, especially in Physics. Less
More informationMultiple View Geometry in Computer Vision
Multiple View Geometry in Computer Vision Prasanna Sahoo Department of Mathematics University of Louisville 1 Basic Information Instructor: Professor Ron Sahoo Office: NS 218 Tel: (502) 852-2731 Fax: (502)
More informationPappus Theorem for a Conic and Mystic Hexagons. Ross Moore Macquarie University Sydney, Australia
Pappus Theorem for a Conic and Mystic Hexagons Ross Moore Macquarie University Sydney, Australia Pappus Theorem for a Conic and Mystic Hexagons Ross Moore Macquarie University Sydney, Australia Pappus
More informationOn the structure of the directions not determined by a large affine point set
On the structure of the directions not determined by a large affine point set Jan De Beule, Peter Sziklai, and Marcella Takáts January 12, 2011 Abstract Given a point set U in an n-dimensional affine space
More informationMagic Circles in the Arbelos
The Mathematics Enthusiast Volume 7 Number Numbers & 3 Article 3 7-010 Magic Circles in the Arbelos Christer Bergsten Let us know how access to this document benefits you. Follow this and additional works
More informationGeometric algebra: a computational framework for geometrical applications (part II: applications)
Geometric algebra: a computational framework for geometrical applications (part II: applications) Leo Dorst and Stephen Mann Abstract Geometric algebra is a consistent computational framework in which
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x
More informationLINES IN P 3. Date: December 12,
LINES IN P 3 Points in P 3 correspond to (projective equivalence classes) of nonzero vectors in R 4. That is, the point in P 3 with homogeneous coordinates [X : Y : Z : W ] is the line [v] spanned by the
More informationAlgebra Workshops 10 and 11
Algebra Workshops 1 and 11 Suggestion: For Workshop 1 please do questions 2,3 and 14. For the other questions, it s best to wait till the material is covered in lectures. Bilinear and Quadratic Forms on
More informationUNIT-I CURVE FITTING AND THEORY OF EQUATIONS
Part-A 1. Define linear law. The relation between the variables x & y is liner. Let y = ax + b (1) If the points (x i, y i ) are plotted in the graph sheet, they should lie on a straight line. a is the
More informationApplications of Geometric Algebra in Computer Vision
Applications of Geometric Algebra in Computer Vision The geometry of multiple view tensors and 3D-reconstruction Christian B.U. Perwass January 2000 To my parents Die menschliche Vernunft hat das besondere
More informationGrade: 10 Mathematics Olympiad Qualifier Set: 2
Grade: 10 Mathematics Olympiad Qualifier Set: 2 ----------------------------------------------------------------------------------------------- Max Marks: 60 Test ID: 22101 Time Allotted: 40 Mins -----------------------------------------------------------------------------------------------
More informationForum Geometricorum Volume 9 (2009) FORUM GEOM ISSN On Three Circles. David Graham Searby
Forum Geometricorum Volume 9 (2009) 181 193. FORUM GEOM ISSN 1534-1178 On Three Circles David Graham Searby Abstract. The classical Three-Circle Problem of Apollonius requires the construction of a fourth
More informationW.K. Schief. The University of New South Wales, Sydney. [with A.I. Bobenko]
Discrete line complexes and integrable evolution of minors by W.K. Schief The University of New South Wales, Sydney ARC Centre of Excellence for Mathematics and Statistics of Complex Systems [with A.I.
More informationThe Spectral Basis of a Linear Operator
The Spectral Basis of a Linear Operator Garret Sobczyk Universidad de Las Americas-P Cholula, Mexico http://www.garretstar.com Thursday Jan. 10, 2013, 2PM AMS/MAA Joint Math Meeting San Diego Convention
More informationFILL THE ANSWER HERE
HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP. If A, B & C are matrices of order such that A =, B = 9, C =, then (AC) is equal to - (A) 8 6. The length of the sub-tangent to the curve y = (A) 8 0 0 8 ( ) 5 5
More informationLECTURE 10: THE PARALLEL TRANSPORT
LECTURE 10: THE PARALLEL TRANSPORT 1. The parallel transport We shall start with the geometric meaning of linear connections. Suppose M is a smooth manifold with a linear connection. Let γ : [a, b] M be
More informationCharacterizations of the finite quadric Veroneseans V 2n
Characterizations of the finite quadric Veroneseans V 2n n J. A. Thas H. Van Maldeghem Abstract We generalize and complete several characterizations of the finite quadric Veroneseans surveyed in [3]. Our
More informationA plane in which each point is identified with a ordered pair of real numbers (x,y) is called a coordinate (or Cartesian) plane.
Coordinate Geometry Rene Descartes, considered the father of modern philosophy (Cogito ergo sum), also had a great influence on mathematics. He and Fermat corresponded regularly and as a result of their
More informationGeometric Algebra with Conzilla Building a Conceptual Web of Mathematics
CID-201 ISSN 1403-0721 Department of Numerical Analysis and Computer Science KTH Geometric Algebra with Conzilla Building a Conceptual Web of Mathematics Mikael Nilsson Master Thesis in Mathematics CID,
More informationCalgary Math Circles: Triangles, Concurrency and Quadrilaterals 1
Calgary Math Circles: Triangles, Concurrency and Quadrilaterals 1 1 Triangles: Basics This section will cover all the basic properties you need to know about triangles and the important points of a triangle.
More informationk times l times n times
1. Tensors on vector spaces Let V be a finite dimensional vector space over R. Definition 1.1. A tensor of type (k, l) on V is a map T : V V R k times l times which is linear in every variable. Example
More informationLinear Analysis Lecture 5
Linear Analysis Lecture 5 Inner Products and V Let dim V < with inner product,. Choose a basis B and let v, w V have coordinates in F n given by x 1. x n and y 1. y n, respectively. Let A F n n be the
More informationLax embeddings of the Hermitian Unital
Lax embeddings of the Hermitian Unital V. Pepe and H. Van Maldeghem Abstract In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital U of PG(2, L), L a quadratic
More informationGeometric Algebra for Computer Science Answers and hints to selected drills and exercises. Leo Dorst, Daniel Fontijne and Stephen Mann
Geometric Algebra for Computer Science Answers and hints to selected drills and exercises Leo Dorst, Daniel Fontijne and Stephen Mann October 28, 2010 2 FINAL October 28, 2010 When we wrote the drills
More informationUNIVERSAL GEOMETRIC ALGEBRA
SIMON STEVIN, A Quarterly Journal of Pure and Applied Mathematics Volume 62 (1988), Number 3 4(September December 1988) UNIVERSAL GEOMETRIC ALGEBRA David Hestenes The claim that Clifford algebra should
More informationAlgebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?
Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials
More informationMA30231: Projective Geometry
MA30231: Projective Geometry Fran Burstall with corrections by James Williams Phil Smith Derek Moniz Ioanna Stylianou Alex Abboud Pooja Khatri Chloe Webbe Hugo Govett Daniel Hurst Tyrah Sanchez Dan Gardham
More informationChapter 1. Synopsis of Geometric Algebra. c D. Hestenes 1998
c D. Hestenes 1998 Chapter 1 Synopsis of Geometric Algebra This chapter summarizes and extends some of the basic ideas and results of Geometric Algebra developed in a previous book NFCM (New Foundations
More information(x, y) = d(x, y) = x y.
1 Euclidean geometry 1.1 Euclidean space Our story begins with a geometry which will be familiar to all readers, namely the geometry of Euclidean space. In this first chapter we study the Euclidean distance
More informationMATH PROBLEM SET 6
MATH 431-2018 PROBLEM SET 6 DECEMBER 2, 2018 DUE TUESDAY 11 DECEMBER 2018 1. Rotations and quaternions Consider the line l through p 0 := (1, 0, 0) and parallel to the vector v := 1 1, 1 that is, defined
More informationIYGB. Special Paper U. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas
IYGB Special Paper U Time: 3 hours 30 minutes Candidates may NOT use any calculator Information for Candidates This practice paper follows the Advanced Level Mathematics Core Syllabus Booklets of Mathematical
More information